The Dimension-Shift Category and Its Mellin-Gamma Representation

We define a thin category $\mathrm{Dim}^+$ of dimension shifts and a category $\mathrm{RadMeas}$ of positive Radon measures with Radon--Nikodym density morphisms. We classify scaling-covariant functors $\mathrm{Dim}^+\to\mathrm{RadMeas}$ whose morphisms are given by homogeneous densities. Gaussian normalization selects a unique functor with values $d\mu_x(u)=\frac{\pi^{x/2}}{\Gamma(x/2)}u^{x/2-1}\,du.$ Its morphism component yields the radial-integration transport $R(x,r)=\frac{\pi^r\Gamma(x/2)}{\Gamma(x/2+r)},$ while the unit-interval observable recovers the Euclidean ball-volume formula $V(x)=\frac{\pi^{x/2}}{\Gamma(x/2+1)}.$ The two transports differ by the multiplicative coboundary of $\beta(x)=x$, identified with the categorical dimension of the standard object in Deligne's interpolation category $\mathrm{Rep}(O_t)$.